Solid mechanics

Solid mechanics Define the terms

Stress Deformation Strain Thermal strain Thermal expansion coefficient

Appropriately relate various types of stress to the correct corresponding strain using elastic theory

Give qualitative descriptions of how intrinsic stress can form within thin films

Calculate biaxial stress resulting from

thermal mismatch in the deposition of thin films

Calculate stresses in deposited thin films using the disk method

Why?

Solid mechanics...

Why?Why?

Why?

A bi-layer of TiNi and SiO2. (From Wang, 2004)

Why is this thing bent?

Thermal actuator produced by Southwest Research Institute

And these?

Why?

A simple piezoelectric actuator design: An applied voltage causes stress in the piezoelectric thin film stress causing the membrane to bend

Membrane is piezoresistive; i.e., the electrical resistance changes with deformation.

Adapted from MEMS: A Practical Guide to Design, Analysis, and Applications, Ed. Jan G. Korvink and Oliver Paul, Springer, 2006

Why?

Hot arm actuator

i

+e-

Joule heating leads to different rates of thermal expansion, in turn causing stress and deflection.

+e-

Zap it with a voltage here… How much does it move here?

ω

Stress and strain

Normal stress

w

w

t

A = w·t

σ = — = —

PP

P

A

P

wt

[F ]

[A ]

[F ]

[L ]2

Dimensions Typical units

N

m 2Pa

δ

ε = — δ

L

L

Dimensions Typical units[L ]

[L ]μ-strain = 10-6

Normal strain

(dimensionless)

Elasticity

How are stress and strain related to each other?

σ

εδ

L

P

PX fracture

X fractureplastic (permanent) deformation

elastic (permanent) deformation

E

E σ = εE

Young’s modulus(Modulus of elasticity,

Elastic modulus)

brittleductile

F = kx

Elasticity

Strain in one direction causes strain in other directions

εy = εx

x

y

-ν

Poisson’s ratio

Stress generalized

Stress is a surface phenomenon.

y

x

z

σy

σx

σz

τxy

τxz

τyx

τyz

τzyτzx

σ : normal stressForce is normal to surfaceσx stress normal to x-surface

τ : shear stressForce is parallel to surfaceτxy stress on x-surface in y-direction

ΣF = 0, ΣMo = 0

τxy = τyx τyz = τzy τzx = τxz

Strain generalized

Essentially, strain is just differential deformation.

Δx

Δy Deforms

Δx + dΔx

Δy + dΔy

dx

dy

u: displacement

ux ux + dx

dux

duy

dxdu

dydu yx

xydxdux

x 21

+=

Break into two pieces:

uniaxial strain shear strain

θ1

θ2 Shear strain is strain with no volume change.

Relation of shear stress to shear strain

Just as normal stress causes uniaxial (normal) strain, shear stress causes shear strain.

dux

duy

θ1

θ2τxy = γxy τxy

τyx

τxy

τyx

G

shear modulus

Magic Algebra Box)1(2

EG

• Si sabes cualquiera dos de E, G, y ν, sabes el tercer.

• Limits on ν:

0 < ν < 0.5ν = 0.5

incompressible

Generalized stress-strain relations

The previous stress/strain relations hold for either pure uniaxial stress or pure shear stress. Most real deformations, however, are complicated combinations of both, and these relations do not hold

Deforms

εx = [ ] + [ ] + [ ]

x normal strain due to x normal

stress

x normal strain due to y normal

stress

x normal strain due to z normal

stress

Ex

Ey

Ez

-ν -ν τxy = G γxy

Generalized Hooke’s Law

εx =

εy =

εz =

γxy =

γyz =

γzx =

zyxE

1

xzyE

1

yxzE

1

xyG1

yzG1

zxG1

For a general 3-D deformation of an isotropic material, then

Generalized Hooke’s Law

Special cases

• Uniaxial stress/strain

σ = Eε

• No shear stress, todos esfuerzos normales son iguales

σx = σy = σz = σ = K•(ΔV/V)

• Biaxial stressStress in a plane, los dos esfuerzos normales son iguales

σx = σy = σ = [E / (1 - ν)] • ε

bulk modulus

biaxial modulus

volume strain

Elasticity for a crystalline silicon

The previous equations are for isotropic materials. Is crystalline silicon isotropic?

E Cij Compliance coefficients

yz

xz

xy

z

y

x

yz

xz

xy

z

y

x

CC

CCCCCCCCCC

44

44

44

111212

121112

121211

000000000000000000000000

yz

xz

xy

z

y

x

yz

xz

xy

z

y

x

CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC

666564636261

565554535251

464544434241

363534333231

262524232221

161514131211

yz

xz

xy

z

y

x

yz

xz

xy

z

y

x

For crystalline siliconC11 = 166 GPa, C12 = 64 GPa and C44 = 80 GPa

Te toca a ti

Assuming that elastic theory holds, choose the appropriate modulus and/or stress-strain relationship for each of the following situations. 1. A monkey is hanging on a rope, causing it to stretch. How do you model the

deformation/stress-strain in the rope?

2. A water balloon is being filled with water. How do you model the deformation/stress-strain in the balloon membrane?

3. A nail is hammered into a piece of plywood. How do you model the deformation/stress-strain in the nail?

4. A microparticle is suspended in a liquid for use in a microfluidic application, causing it to compress slightly. How do you model the deformation/stress-strain in the microparticle?

5. A thin film is deposited on a much thicker silicon wafer. How do you model the deformation/stress-strain in the thin film?

6. A thin film is deposited on a much thicker silicon wafer. How do you model the deformation/stress-strain in the wafer?

7. A thin film is deposited on a much thicker glass substrate. How do you model the deformation/stress-strain in the glass substrate?

Uniaxial stress/strain

Biaxial stress/strain

Uniaxial stress/strain

Use bulk modulus (no shear, all three normal stresses the same)


Anisotropic stress/strain (Using Cij compliance coefficients)

Generalized Hooke’s Law. I.e., ε = (1/E)(σx – ν(σy + σz)) etc.

Thermal strain

Most things expand upon heating, and shrink upon cooling.

Thermal Expansion

dTd T

T

Thermal expansion coefficient

Notes:

• αT ≈ constant ≠ f(T)

ε(T) ≈ ε(T0) + αT (T-T0)

• Thermal strain tends to be the same in all directions even when material is otherwise anisotropic.

If no initial strain

δ(T) = αT (T-T0)

Solid mechanics of thin films

Adhesion

Ways to help ensure adhesion of deposited thin films:• Ensure cleanliness• Increase surface roughness• Include an oxide-forming

element in between a metal deposited on oxide

Stress in thin films

positive (+) Negative (-)Tension headacheTension Compression


Intrinsic stressAlso known as growth stresses, these develop during as the film is being formed.

Extrinsic stress

These stresses result from externally imposed factors. Thermal stress is a good example.

Two types of stress

Doping

SputteringMicrovoidsGas entrapment

Polymer shrinkage

Thermal stress in thin films

Consider a thin film deposited on a substrate at a deposition temperature, Td. (Both the film and the substrate are initially at Td.)

Initially the film is in a stress free state.

The film and substrate are then allowed to cool to room temperature, Tr

Since the two materials are hooked together, they both experience the same ____________ as they cool.

εboth = εsubstrate or εfilm ?

εsubstrate =

αT,s(Tr - Td)

εfilm =

αT,f (Tr - Td)

εmismmatch = αT,s(Tr - Td) - αT,f (Tr - Td)

= (αT,f - αT,s)(Td - Tr)

strain

=

substrate

thin film deposited at Td

both cooled to Tr

+ εmismmatch

Thermal stress in thin films

How would you relate σmismatch to εmismatch?

σmismatch = [E / (1-ν)]·εmismatch


= [E / (1-ν)]·(αT,f - αT,s)(Td - Tr)

If αT,f > αT,s σmismatch = (+) or (-) Film is in ___________________. If αT,f < αT,s σmismatch = (+) or (-) Film is in ___________________.

Sacrificial layer

Initially stress free cantilever

Thin film

tension

compression

σmismatch > 0 σmismatch < 0


(a) (b)

(a) Stress in SiO2/Al cantilevers (b) Stress in SiO2/Ti cantilevers

[From Fang and Lo, (2000)]

Compression or tension? Compression or tension?

αT,Al >, <, = αT,SiO2 ? αT,Ti >, <, = αT,SiO2 ?


(a) (b)

(a) Stress in SiO2/Al cantilevers (b) Stress in SiO2/Ti cantilevers

[From Fang and Lo, (2000)]

How were these fabricated?

Te toca a ti

Show that the biaxial modulus is given by

E/(1 – ν)

Pistas:• Remember what the assumptions for “biaxial” are.• In thin films you can always find one set of x-y axes for which

there is only σ and no τ.

Measuring thin film stress

Stressed wafer (after thin film)

R

Assumptions: The film thickness is uniform and small

compared to the wafer thickness. The stress in the thin film is biaxial and

uniform across it’s thickness. Ths stress in the wafer is equi-biaxial (biaxial at

any location in the thickness). The wafer is unbowed before the addition of

the thin film. Wafer properties are isotropic in the direction

normal to the film. The wafer isn’t rigidly attached to anything

when the deflection measurement is made.

Unstressed wafer (before thin film)

RtTE61

2

R = _________________________, T = _________________________ and t = _________________________.

The disk method

radius of curvature

Biaxial modulus of the wafer

wafer thickness

thin film thicknessstrain at

wafer/film interface

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Solid mechanics